Theorem univ 4527

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3851	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3862	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4265	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2229	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1373  Vcvv 2773  𝒫 cpw 3617  ∪ cuni 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-uni 3853

This theorem is referenced by: (None)